1987 Nishiki Ariel Needs Help

I got this bike in the divorce settlement with my ex. It was hers. I had the Colorado. At any rate, It's been passed to my son who uses it as a commuter. He's kinda fallen in love with the thing and when I suggested we buy him new commuter, he said, " Why don't we just fix up the Nishiki? That frame still has a lot of miles in it." He rides the thing to and from school everyday, basically 7 miles each way. So we are going to save the Ariel.
As I'm looking at it, itneeds new rear wheel...probably a front to, but the one on there now is original.
It needs a new bottom bracket and crankset. With the new wheels we will probably do some new brakes as well.
I'm looking for recommendations on where to find parts for the poor thing.
Any help is appreciated.

Any normal kid would leap at the opportunity to get a new bike, your son is weird. (I mean that in a good way of course!)

First up, work with what's there if you can. As in, clean it, service it and replace only what you need to.

As for wheels, what's wrong with them? If the hubs are spinning freely then you can get new rims laced to those hubs. Though quality rim brake rims are getting harder to find now that disc brakes are the norm (disc brake wheels don't have a braking surface or even a sidewall for the rim brake pad).

Otherwise, online auction and sale sites can be okay if you're careful not to buy someone else's woes or something that's in equal or worse condition.

What's wrong with the cranks and BB? Is the BB a sealed unit or can it be serviced (ie: cup and cone bearing). You have some options, one of which is simply a new Shimano SLX or Deore crank and external BB setup which runs pretty cheap and is pretty bomb proof. On an older bike it will stick out like canine spheroids but it's a more foolproof option than hunting down a BB and hoping to get the axle length to work with whatever crank you end up getting. But there's nothing to stop you buying a square taper BB to match the shell and axle of the existing one and keeping the cranks (assuming the chainrings aren't flogged out).

Brakes, get some new pads and clean up what's there. If you get v-brakes then that will mean new levers because the cable pull is different between cantilever brakes and v brakes.

Factor in some new cables and outers into the mix and a bit of grease, both elbow and the other sort.

Where to buy parts? Bike shop. Not only will they be able to source parts but bike shops will look at the bike and be able to offer more accurate advice as well as any service or repair that you might not be able to do.

Yep, agreed. The stated dollar amount to spend is in the neighborhood of 200.00 for now. Tires, shifters, seat, frame, cockpit and pedals are good to go. Maybe just needs some brake pads that will come when the new wheel is sourced. Currently, I just rode it last weekend, it's not very safe for me. Son, on the other hand, seems to have no fear, just wants the crank to be solid and for it to stop making noise. It makes a lot of noise. He is an animal on that thing, I must say. A lesser bike would have thrown itself into a ditch or something to avoid the hard work he puts on that thing.

Yep, agreed. The stated dollar amount to spend is in the neighborhood of 200.00 for now. Tires, shifters, seat, frame, cockpit and pedals are good to go. Maybe just needs some brake pads that will come when the new wheel is sourced. Currently, I just rode it last weekend, it's not very safe for me. Son, on the other hand, seems to have no fear, just wants the crank to be solid and for it to stop making noise. It makes a lot of noise. He is an animal on that thing, I must say. A lesser bike would have thrown itself into a ditch or something to avoid the hard work he puts on that thing.

Cranks themselves don't make noise unless they're cracked or the interface between the crank and the BB spindle is dry or worn to the point that there is play. The noise is likely to be the bottom bracket. If it's a cup and cone then you can service it, otherwise if its a sealed unit then replace it. Square taper BBs are still out there. just make sure you get the right shell width and axle length by measuring what you have now.